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Homework answers / question archive / A 1-year European call option on a stock with a continuous dividend rate of 2%
A 1-year European call option on a stock with a continuous dividend rate of 2%. You are given: The current price is 50 and the strike price is 55. At the end of a year, the stock price will be either 40 or 60. The risk-free rate is 5% Calculate the option premium.
Su = price in the up state = 60; Sd = price in the down stgatge = 40; S0 = Current price = 50; Strike price, K = 55
Continuous div rate, ? = 2%; Risk free rate = r = 5%; Time to maturity, t = 1 year
up factor, u = Su/S0 = 60/50 = 1.2
down factor, d = Sd/S0 = 40/50 = 0.8
Risk neutral probability of an up state = p = [e(r - ?) x t - d] / (u - d) = [e(5% - 2%) x 1 - 0.8] / (1.2 - 0.8) = 0.5761
Payoff of the call option in up state = Cu = max (Su - K, 0) = max (60 - 50, 0) = 10
Payoff of the call option in down state = Cd = max (Sd - K, 0) = max (40 - 50, 0) = 0
Hence, the option premium = [p x Cu + (1 - p) x Cd] x e-r x t = [0.5761 x 10 + (1 - 0.5761) x 0] x e-5% x 1 = 5.48
